A key issue is whether the branch of travelling wave solutions which emerge from the neutral curve is subcritical and so exist down to some saddle node at Weissenberg number \(Wi_{sn}\)  below the critical value \(Wi_c\) , thereby potentially connecting the instability to either EIT and/or ET in parameter space.
[1]} demonstrated the existence of substantial subcriticality albeit at \(Re=60\)  (and \(\beta =0.9\) ) where \(Wi_{sn}=8.8\)  is much lower than \(Wi_c=26.7\) . Despite EIT not existing at this low \(Re\) , the upper branch travelling waves found there clearly resembled the `arrowhead' states found in the simulations of [2]} at \(Re=1000\)  when EIT was annealed by increasing the elasticity. Weakly nonlinear analysis (in the channel by [3]} and pipe flow by [4]}) has confirmed the general subcritical nature of the instability but can not give global information about how far \(Wi_{sn}(Re,\beta )\)  is below \(Wi_c(Re,\beta )\) .
Our purpose here is to answer this by performing an investigation using branch continuation to track where the travelling waves exist in \((Wi,Re,\beta )\) -parameter space.
This turns out to be feasible, as a branch continuation procedure based on solving an algebraic set of equations derived directly from the governing equations is much more efficient than branch continuing using a DNS code as done in [1]}. There are two reasons for this. Firstly, the travelling wave is highly symmetric: it is 2-dimensional and has a symmetry around the channel's midplane. Secondly, far fewer degrees of freedom are needed to resolve the flow algebraically compared to the number needed to keep a time-stepping code stable. For example, the algebraic formulation needs only \(\approx 50\)  Chebyshev modes in the cross-stream direction for convergence at the parameters considered while the DNS code needs \(\approx 128\)  modes to stay time-stable.
